special primes in sets of five

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• Im was looking at metric topology as applied to primes and thought of doing a kind of sequential measure using matrices. I made up two matrices of primes A and
Message 1 of 1 , Jul 1, 2005
Im was looking at metric topology as applied to primes and thought of doing
a kind of sequential measure using matrices.
I made up two matrices of primes
A and B as sequenctial:
M.A=B
I then solved for M as:
M=B.A^(-1)
Looking at the 2by2 versions:
A={{Prime[n-1], Prime[n]},
{Prime[n],Prime[n+1]}
B={{Prime[n], Prime[n+1]},
{Prime[n+1],Prime[n+2]}
measure=Det[B.A(-1)]
I get it alternating in sign, but nothing really I can use,
but the 3by3 version has points where the bottom is zero or M is
simgular:( Det[A]=0)
(((Prime[1 + n]^3 + Prime[-1 + n]*Prime[2 + n]^2 + Prime[n]^2 Prime[3 +
n] - Prime[1 + n]*((2* Prime[n]*Prime[2 + n] + Prime[1) + n]*Prime[3 +
n])))))=0
In the first 2000 primes it happens only 5 times:
{{23, 29, 31, 37, 41},
{2237, 2239, 2243, 2251, 2267},
{3433, 3449, 3457, 3461, 3463},
{6827, 6829, 6833, 6841, 6857},
{17387, 17389, 17393, 17401, 17417}}
A = {{Prime[n - 1], Prime[n], Prime[n + 1]},
{Prime[n], Prime[n + 1], Prime[n + 2]},
{Prime[n + 1], Prime[n + 2], Prime[n + 3]}}
B={{Prime[n], Prime[n + 1], Prime[n + 2]},
{Prime[n + 1], Prime[n + 2], Prime[n + 3]},
{Prime[n + 2], Prime[n + 3], Prime[n + 4]}}
M=B.A^(-1) is too complex to show but the term that goes to zero comes
from the inversion of A ( Det[A]).
In the two by two system it never is singular,
so I seem to have discovered an effect of some sort.
Det[M]=Det[B]/Det[A]=Infinity
Or
1/Det[M]=0
Looking at Det[B]=0 I get the same sets of five numbers as well, as I
should.
These sets of five seem to be little isolated islands of primes.
The question is:
does a determinant measure of this kind exist for 4by4 and 5by5 matrices
as well?

%I A000001
%S A000001 23, 29, 31, 37, 41, 2237, 2239, 2243, 2251, 2267, 3433, 3449, 3457, 3461,
3463, 6827, 6829, 6833, 6841, 6857, 17387, 17389, 17393, 17401, 17417, 23173,
23189, 23197, 23201, 23203, 26083, 26099, 26107, 26111, 26113, 31123, 31139,
31147, 31151, 31153, 37307, 37309, 37313, 37321, 37337, 39343, 39359, 39367,
39371, 39373, 43397, 43399, 43403, 43411, 43427, 58907, 58909, 58913, 58921,
58937, 65837, 65839, 65843, 65851, 65867, 89597, 89599, 89603, 89611, 89627
%N A000001 a sequence of sets of five primes whose 3by3 determinants are zero.
%C A000001 In sets of five each:
{{23, 29, 31, 37, 41},
{2237, 2239, 2243, 2251, 2267},
{3433, 3449, 3457, 3461, 3463},
{6827, 6829, 6833, 6841, 6857},
{17387, 17389, 17393, 17401, 17417},
{23173, 23189, 23197, 23201, 23203},
{26083, 26099, 26107, 26111, 26113},
{31123, 31139, 31147, 31151, 31153},
{37307, 37309, 37313, 37321, 37337},
{39343, 39359, 39367, 39371, 39373},
{43397, 43399, 43403, 43411, 43427},
{58907, 58909, 58913, 58921, 58937},
{65837, 65839, 65843, 65851, 65867},
{89597, 89599, 89603, 89611, 89627}}
%F A000001 A = {{Prime[n - 1], Prime[n], Prime[n + 1]},
{Prime[n], Prime[n + 1], Prime[n + 2]},
{Prime[n + 1], Prime[n + 2], Prime[n + 3]}}
a(m) =if Det[A]=0 then {{Prime[n - 1], Prime[n], Prime[n + 1], Prime[n + 2], Prime[n + 3]}}
%t A000001 a = Flatten[Table[If[(((Prime[1 + n]^3 + Prime[-1 + n]*Prime[2 + n]^2 + Prime[n]^2 Prime[3 + n] - Prime[1 + n]*((2* Prime[n]*Prime[2 + n] + Prime[1) + n]*Prime[3 + n]))))) == 0, {{Prime[n - 1], Prime[n], Prime[n + 1], Prime[n + 2], Prime[n + 3]}}, {}], {n, 2, 10000}], 1]
Flatten[a]
%O A000001 1
%K A000001 ,nonn,
%A A000001 Roger L. Bagula (rlbagulatftn@...), Jul 01 2005
RH
RA 69.108.48.187
RU
RI

--
Roger L. Bagula { email: rlbagula@... or rlbagulatftn@... }